Soliton multidimensional equations and integrable evolutions preserving Laplace’s equation
نویسنده
چکیده
The KP equation, which is an integrable nonlinear evolution equation in 2 + 1, i.e., two spatial and one temporal dimensions, is a physically significant generalization of the KdV equation. The question of constructing an integrable generalization of the KP equation in 3+1, has been one of the central open problems in the field of integrability. By complexifying the independent variables of the KP equation, I obtain an integrable nonlinear evolution equation in 4 + 2. The requirement that real initial conditions remain real under this evolution, implies that the dependent variable satisfies a nonlinear evolution equation in 3 + 1 coupled with Laplace’s equation. A reduction of this system of equations to a single equation in 2 + 1 contains as particular cases certain singular integro-differential equations which appear in the theory of water waves. © 2007 Published by Elsevier B.V.
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